Active return is a portfolio’s return minus the return of its chosen benchmark, measured over the same period. It tells an investor how far the portfolio finished ahead of or behind that reference. A profitable fund can have negative active return if its benchmark did better; a fund can outperform while losing money if its benchmark lost more.
For simple returns and , the arithmetic active return is
The benchmark is part of the definition. Outperformance against a broad equity index and outperformance against a cash hurdle answer different investment questions. Choose a reference that reflects the mandate, and identify it when reporting the result. “Excess return” can also mean return above the risk-free rate, so specify what has been subtracted.
Three percentage points, or 2.857%?
Suppose a portfolio and benchmark earn nominal USD total returns of 8% and 5% over one matched period. Both include reinvested distributions; this synthetic example has no costs, taxes or external cash flows.
The portfolio’s arithmetic active return is 3 percentage points. Starting each investment with USD100 leaves USD108 in the portfolio and USD105 in the benchmark: an extra USD3 per USD100 initially invested.
There is another useful comparison: how much larger is the portfolio’s ending wealth than the benchmark’s ending wealth?
This is often called geometric excess return or wealth-relative return. Its denominator is benchmark ending wealth. Arithmetic active return uses a difference between returns. The two results describe the same USD3 gap using different denominators; neither should be silently substituted for the other. The wealth-relative formula requires when interpreted relative to positive benchmark wealth.
Original synthetic one-period USD total-return ledger. Both investments start with USD100, with no costs, taxes or external cash flows. The same USD3 gap produces different numerical measures because their denominators differ.
Across periods, compound the two investments first
Continue the example for a second consecutive period: the portfolio returns −2% and the benchmark −1%. Keep the same conventions and reinvest all wealth.
| Measurement | Portfolio | Benchmark | Arithmetic active return |
|---|---|---|---|
| First-period return | 8% | 5% | 3 percentage points |
| Second-period return | −2% | −1% | −1 percentage point |
| Cumulative return over both periods | 5.84% | 3.95% | 1.89 percentage points |
| Ending wealth from USD100 | USD105.84 | USD103.95 | — |
The portfolio compounds to ; the benchmark compounds to . Their cumulative return difference is 1.89 percentage points, although the two period-by-period active returns add to 2 percentage points. Different second-period wealth bases create the discrepancy.
For a horizon of consecutive sub-periods, the cumulative arithmetic difference is
Wealth-relative returns can instead be linked multiplicatively:
provided the benchmark wealth denominators remain positive. An arithmetic mean of periodic active returns serves a different purpose, such as the numerator of an information ratio; it is not the cumulative return difference. If external cash flows occur, measure each investment’s returns consistently before comparing them. The Risk-Adjusted Performance Measures lesson develops time-weighted chaining and performance ratios; a public preview is available and the full lesson requires membership.
What the comparison must hold fixed
Use the same start and end dates, valuation timing and return frequency. Express both returns in the same reporting currency, and distinguish nominal from inflation-adjusted returns. An unhedged foreign-currency index and a currency-hedged mandate may capture different exposures even when both are reported in USD.
Compare total returns on both sides. A portfolio return including dividends compared with a price-only index mixes income with relative investment performance. State distribution-reinvestment and relevant withholding-tax conventions.
Also identify the cost basis. Portfolio returns after management fees, transaction costs and financing charges answer a different question from returns before those charges. A published index may exclude implementation costs; this can be a useful opportunity-cost comparison, but it does not describe the net return of an investable tracker. Identify any benchmark fee adjustment rather than assuming both series deduct the same charges.
The GIPS benchmark guidance for firms, revised July 2023, addresses matched periods, currency, total returns and benchmark cost disclosures within GIPS reporting. These conventions make the subtraction interpretable; performing the subtraction alone cannot repair an unsuitable benchmark.
Active return, alpha and tracking error
Alpha, in risk-adjusted performance evaluation, asks what remains after accounting for specified systematic exposures. Beating an index can reflect a larger market exposure, a rewarded style tilt, luck or skill. The realised difference alone does not distinguish them. Some practitioners call benchmark-relative return “alpha”; state the convention, especially when comparing it with a regression intercept.
Tracking error measures the variability of active returns, rather than the level of one observation. Historical tracking error uses a series of matched return differences; forward-looking tracking error uses a risk model. One period’s outperformance is insufficient to establish either a historical volatility or persistent skill.
For a one-period fully invested portfolio and benchmark holding the same asset universe, beginning weights give a useful bridge from holdings to performance:
The differences are active weights. This identity assumes common asset total returns, no intervening external cash flows or rebalancing, and no separate unallocated costs. It shows how overweights and underweights generate the realised difference. The Mean-Variance Optimisation lesson develops active weights and modelled active risk; it has a public preview, with full access requiring membership. Selecting those weights is an investment decision. Establishing that they deliver repeatable risk-adjusted value requires more evidence than positive active return.
Sources
- Edward E. Qian, Ronald H. Hua and Eric H. Sorensen, Quantitative Equity Portfolio Management: Modern Techniques and Applications, Chapman & Hall/CRC (2007), §2.2.4, pp. 34–35 — benchmark-relative return, active weights and active risk. Numerical ledgers and the figure here are original.
- CFA Institute, Analysis of Active Portfolio Management — public summary on benchmark-relative value added, beginning active weights and active risk.
- CFA Institute, Guidance Statement on Benchmarks for Firms, effective April 2021, revised July 2023, pp. 2, 5–10 and 21 — matched periods, benchmark selection and costs, currency, and total-return comparisons.